Mean-Field and Classical Limit for the N-Body Quantum Dynamics with Coulomb Interaction

This paper proves the validity of the joint mean-field and classical limit of the bosonic quantum N-body dynamics leading to the pressureless Euler-Poisson system for factorized initial data whose first marginal has a monokinetic Wigner measure. The interaction potential is assumed to be the repulsive Coulomb potential. The validity of this derivation is limited to finite time intervals on which the Euler-Poisson system has a smooth solution that is rapidly decaying at infinity. One key ingredient in the proof is an inequality taken from S. Serfaty (Duke Math. J. 169 (2020), 2887–2935)).